Is there a way to keep a private prayer list on one machine and not have it sync to the other one when they both have the same accounts?
Not individually. as of right now the only way not to sync one thing is to not sync anything (e.g. completely offline).
If there is more than one user you need separate Logos accounts!
You could keep Libronix (Logos 3) around for this. It doesn't sync. It had the Prayer List feature, and I seem to recall some people saying its implementation of it was better than L4's, though that might have been rectified already. I haven't played around with it much.
